Term Rewriting System R:

   [x]
   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   FAC(s(x)) -> FAC(p(s(x)))
   FAC(s(x)) -> P(s(x))
   P(s(s(x))) -> P(s(x))

Furthermore, R contains two SCCs.


SCC1:

P(s(s(x))) -> P(s(x))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(P(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   P(s(s(x))) -> P(s(x))



 This transformation is resulting in no new subcycles.


SCC2:

FAC(s(x)) -> FAC(p(s(x)))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(FAC(x_1)) = 1 + x_1
POL(0) = 0
POL(p(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   p(s(0)) -> 0


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

FAC(s(x)) -> FAC(p(s(x)))

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   FAC(s(x)) -> FAC(p(s(x)))
one new Dependency Pair
is created:


   FAC(s(s(x''))) -> FAC(s(p(s(x''))))

The transformation is resulting in one subcycle:


SCC2.MRR1.Nar1:

FAC(s(s(x''))) -> FAC(s(p(s(x''))))

The Estimated Innermost Dependency Pair Graph for R contains the following SCC: 

FAC(s(s(x''))) -> FAC(s(p(s(x''))))
Rule(s) before reversing: 

   p(s(s(x))) -> s(p(s(x)))
   FAC(s(s(x''))) -> FAC(s(p(s(x''))))


(Re)applying the dependency pair methods for the following TRS:
   s(s(p(x))) -> s(p(s(x)))
   s(s(FAC(x))) -> s(p(s(FAC(x))))
 resulting in one new SCC:

   S(s(p(x))) -> S(x)
SCC2.MRR1.Nar1.Rev1:

S(s(p(x))) -> S(x)

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC2.MRR1.Nar1.Rev1.NOC1:

S(s(p(x))) -> S(x)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(p(x_1)) = 1 + x_1
POL(S(x_1)) = 1 + x_1
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   S(s(p(x))) -> S(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 20.680 seconds.